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A Near-optimum Procedure for Selecting Stations in a Streamgaging Network

by Kenneth J. Lanfear

U.S. Geological Survey Scientific Investigations Rpeort 2005-5001--ONLINE ONLY

The report is available as a pdf.


Table of Contents

ABSTRACT
Introduction
     Federal Interests
     Satisfying Goals
     Finding Basic Feasible Incremental Solutions
     Costs and Benefits
Solution Algorithm
     Alternative solution procedures
     Initial Conditions
Example Problem
     Data Sets
     Expert Rules for Selecting BFIS
     Example Costs and Benefits
     Determining the Example BFIS
     Active Stations - Initial Conditions
     Programming Implementation
     Solution for All Goals
     Is the Solution Optimal?
Earlier Applications
Conclusions
Literature Cited

Figures

Figure 1. Graph of the number of goals satisified as a function of the number of stations added
Figure 2. Map of stations meeting example Federal goals. Active stations are shown in light gray and reactive stations in black.

Tables

Table 1. Criteria for satisfying each type of Federal goal.
Table 2. List of relative location costs assigned to solutions.
Table 3. List of the number of goals of each type and the total number of BFIS's that satisfy each goal type.
Table 4. Percent of each type of Federal goal satisified by currently active stations.
Table 5. Example of how the algorithm can make a non-optimal choice.


ABSTRACT

Two questions are fundamental to Federal government goals for a network of streamgages which are operated by the U.S. Geological Survey: (1) how well does the present network of streamagaging stations meet defined Federal goals and (2) what is the optimum set of stations to add or reactivate to support remaining goals? The solution involves an incremental-stepping procedure that is based on Basic Feasible Incremental Solutions (BFIS’s) where each BFIS satisfies at least one Federal streamgaging goal. A set of minimum Federal goals for streamgaging is defined to include water measurements for legal compacts and decrees, flooding, water budgets, regionalization of streamflow characteristics, and water quality. Fully satisfying all these goals by using the assumptions outlined in this paper would require adding 887 new streamgaging stations to the U.S. Geological Survey network and reactivating an additional 857 stations that are currently inactive.

Introduction

Since 1889, the U.S. Geological Survey (USGS) has operated a streamgaging network to collect information about the Nation’s water resources. It is a multipurpose network funded by the USGS and many other Federal, State, and local agencies. Individual streamgaging stations are supported for specific purposes such as water allocation, reservoir operations, or regulating permit requirements, but the data are used for many other purposes. Thomas and Wahl (1993) surveyed cooperators and data users to identify uses of data in 9 categories. They found that uses of data from a typical streamgaging station fall into an average of 2.6 different data-use categories. The USGS recently examined the network to see how it is meeting Federal goals (U.S. Geological Survey, 1998). The evaluation defined key Federal goals for the network and established a set of quantitative metrics that measure the extent to which those goals are being achieved.

The evaluation technique in this report takes earlier analyses by the U.S. Geological Survey (1998; 1999) of how well the existing network meets goals to the next logical phase of determining how to improve the network by proposing an incremental-stepping procedure to select streamgaging stations that meet additional Federal goals. Key to this technique is finding sets of stations or potential new station locations that efficiently meet individual goals. For any individual goal – for example, knowing the flow of a river at particular location – a limited number of practical configurations of stations can achieve the desired accuracy. By choosing from among limited sets of stations, rather than individual stations, we greatly reduce the number of possible solutions and ensure that the procedure will reach a near-optimum solution.

Federal Interests

Federal interests in streamgaging, which are defined in U.S. Geological Survey (1998; 1999), include supporting Federal programs, resolving disputes among states, and managing Federal lands. These broad interest categories were reduced to specific types of Federal goals with quantitative measures for determining success in meeting the goals. The list below includes important goals representative of some major Federal interests. It does not represent a full compilation of Federal interests but is presented to demonstrate the analysis techniques.

Satisfying Goals

Represent N potential streamgaging stations with the N x 1 vector U, where each ui equals 1 if station i is active and 0 if not active. Using the terminology of Yankowitz (1982), Ut is the “control set” of the streamgaging network at stage t. Every stream reach can have one or more active or inactive stations, or is a candidate for a new station. Thus, there are at least as many possible station locations as there are reaches.

Let the M x 1 vector X represent the “state” of M goals. Each xj represents a single goal. Examples of goals are: a NWS Service Location where streamflow must be determined; a basin for which inflow and outflow must be measured to determine the water budget; a basin within an ecoregion where at least one representative sampling station is required; or a long-term water-quality sampling location which requires estimates of streamflow. Let xj equal 1 if the goal is met, 0 if it is not met. A goal is either met or unmet; there is no partial satisfaction. Goals can be coincident. For example, a NWS Service Location might fall on an interstate crossing. Each, however, would be treated as a separate goal.

Define the “dynamics function,” f, which determines if a goal is met, as equation 1:

Equation 1.

Each fj is determined by examining the network and finding those stations or combinations of stations that would satisfy goal j. Assuming that every goal has at least one solution, then there exists at least one Uj such that fj(Uj) equals 1. Now impose another condition to eliminate solutions that contain redundant stations: discard a Uj if any of its stations can be removed without its dynamics function becoming zero. A Uj that fulfils these two conditions for any goal j is called a Basic Feasible Incremental Solution (BFIS). That is, a set of one or more stations (new, reactivated, or existing) that satisfies a goal, with all stations in the set being necessary to satisfy the goal, is a Basic Feasible Incremental Solution (BFIS) for that goal. A station can belong to more than one BFIS.

Finding Basic Feasible Incremental Solutions

Streamflow information commonly is estimated from nearby stations on the stream network; it is not necessary to have a streamgaging station exactly where you need to know the streamflow. Stations can be located upstream or downstream from the point of interest, and, in many cases, discharge values from multiple stations can be added or subtracted. Streamflow characteristics at ungaged sites can be estimated from regression analysis of nearby gaged sites (Moss and Karlinger, 1974, Moss and others, 1982, Moss and Tasker, 1991).

Determining an appropriate set of existing or new stations for estimating streamflow has been the subject of considerable research. Benson and Matalas (1967) showed how basin characteristics could be used to estimate streamflow characteristics at ungaged sites. Stedenger and Tasker (1985) showed how generalized least squares could provide more accurate parameter estimates. Sharp (1971), looking to optimally place stations to measure the effects of known pollutant sources, developed a method for selecting sampling locations based on the topology of the river network. Sanders and others (1983) applied a similar technique for allocating water-quality sampling stations. Karasev (1968) looked at minimum distance criteria.

With multiple objectives, no single method may suffice for selecting the BFIS. A goal of estimating regional streamflow parameters may require BFIS based on regression analysis. Selecting stations for a flood-warning goal would have to consider the stream network topology and travel times. Moreover, any BFIS does not have to yield the most accurate estimate of streamflow; a BFIS need only provide sufficient accuracy to meet the purposes of the goal. The actual selection method for BFIS can be any combination of regression, analysis of stream network topology, examination of actual field conditions, or expert judgment.

Costs and Benefits

Let C be an N x 1 vector of costs associated with each station. Each ci is the cost of activating and operating station i. For any Ut,

Equation 2.

For benefits, let W be an M x 1 vector representing the benefits of each goal. Each wJ. Then,

Equation 3.

Solution Algorithm

A forward-stepping process adds one BFIS at a time in an efficient manner to find a set of stations that satisfies all goals. The BFIS chosen in each step is that which provides the greatest ratio of incremental benefits to incremental cost.

The solution algorithm adds a BFIS in each step until there are no more goals to satisfy. Choose the optimal BFIS on the basis of its incremental cost because some of the stations in the BFIS may already have been added through other BFIS, and incremental benefits because a BFIS may happen to satisfy additional goals through common stations.

Let {B} be the set of all BFIS for all goals. The problem is to select a subset of {B} that optimally meets all goals. Starting with no selections, we will do this by adding one Bt at each stage, t, such that each Bt fulfils at least one new goal. In calculating benefits, we must account for all additional goals that Bt fulfils and assign the costs of any stations needed to complete Bt that are not already in the solution. The incremental costs and benefits of each stage are computed as follows.

For any Bt added,

Equation 4.

Equation 5.

and

Equation 6.

In moving from stage t-1 to t, the gain, expressed as a benefit-cost ratio will be

Equation 7.

The objective in each step is to select Bk to maximize the benefit-cost ratio of that step,

Equation 8.

This is where the BFIS play an important role. Since, by definition, each BFIS satisfies at least one goal, we are guaranteed that for some number of stages T where T < M,

Equation 9.

and

Equation 10.

This ensures that the algorithm will proceed to a solution that satisfies all goals.

Alternative solution procedures

A simpler procedure for selecting an appropriate solution might be to add one station per step, rather than one BFIS. Maddock (1974) used a technique of removing stations one-by-one in looking at how to reduce the number of stations in a network. Tasker (1986) used a backward-stepping technique to select stations that minimized the average sampling mean-square error of a regional regression equation. The problem with adding one station per step is that, at some step, all remaining goals might require two or more stations each. This will set all incremental benefit-cost ratios maxGt-(t-1) equal to zero before all the goals are satisfied, which will cause the algorithm to stall before it achieves a solution. Tasker (1986) did not face this problem because he considered a single goal involving regional regression. A way around this local minimum is an n-step algorithm – that is, trying all combinations of N stations. This method, however, becomes computationally infeasible with a large number of stations. Selecting from among the BFIS, as proposed here, effectively implements an n-step technique with a variable n being the number of stations in each BFIS.

Another variant would be to allow partial goal solutions. That is, allow f to take on any real number 0 ≤ f ≤ 1. While this eliminates the problem of stalling at local minima, it introduces the possibility of selecting stations that ultimately don’t contribute to satisfying a goal. A solution that is not a union of BFIS’s can not be optimal since it will include stations that contribute nothing to the solution.

The solution can not be solved by linear programming because the BFIS can share stations and, thus, are not independent of each other. Fiering (1965) used nonlinear integer programming to decide which sites in a network should be gaged further. His approach bears many similarities to the one proposed here, but it is based on individual stations, not sets of stations.

Burn and Goulter (1991) and Yang and Burn (1994) used clustering techniques to identify groups of similar stations, then selected a single station from each group. Their method is applicable to regionalization, but does not address the problem of multiple goals.

Initial Conditions

One set of initial conditions is to accept the currently active stations as a given. Goals satisfied by the set of active stations are eliminated from the solution before beginning the iteration procedure. Depending on how successful the active stations are in meeting the goals, this approach can substantially reduce the computation time for selecting BFIS to satisfy the remaining goals. Of course, the selected set of stations will be optimized only for the added goals, not the entire set of goals.

Example Problem

Data Sets

An analysis of this scope would not be possible without a substantial infrastructure of geospatial data sets:

Expert Rules for Selecting BFIS

Many hydrologic judgments can be expressed as rules that apply to a network of streams as represented by a GIS. Rules for finding sets of BFIS can address:

The rules applied for selecting stations that satisfy the Federal goals are summarized in table 1. A team of hydrologists experienced in streamgaging determined each rule. The rules were incorporated into computer code that operated on the data sets to find solutions. Using a geographic information system to view the results, the team modified the rules in several rounds of adjustments until they produced logical, practical alternatives. That is, for any goal, the rules would produce one or more alternatives for streamgaging station locations that would be comparable to the choices made by an experienced hydrologist. In effect, the rules are an expert system for choosing streamgaging solutions. Although these rules are generally applicable for stations in the conterminous United States, no final decision about a particular streamgaging situation should be made without examining local site conditions.

Table 1. Criteria for satisfying each type of Federal goal.

GoalPrinciple Criteria1
Compacts and Decrees Each compact or decree is associated with a specific USGS station.
Current NWS Flood-Forecast Sites Must include 90 to 110 percent of the service location’s drainage area and be within 20 km, measured along the streams, of the service location. A solution may have no more than 2 stations, and each must have a drainage area at least 20 percent the size of the service location.
Accounting-Unit Water Budgets Must include 75-125 percent of the accounting unit drainage, with no more than 25 percent of the drainage outside the accounting unit. Large mainstream rivers flowing through the basin are not included in the totals. Where possible, use only reaches with existing (active or inactive) streamgaging stations, but accept new stations if necessary. If possible, the number of stations in a solution should be limited to 3. However, if no solution is found with 3 or fewer stations, then as many as 4 stations may be accepted.
Regionalization One station for each intersection of ecoregions with accounting units. Each station must have a drainage area of less than 100 mi2, or 500 mi2 if it is a Hydroclimatic Data Network (HCDN) station (Slack and Landwehr, 1992), and the drainage must be entirely within the ecoregion-accounting unit intersection.
Quality-Impaired Watersheds Must include at least 20 percent of the cataloging unit drainage, with no more than 20 percent of the drainage outside the cataloging unit. Large mainstem rivers flowing through the watershed are not included in the totals. Where possible, use only reaches with existing (active or inactive) streamgaging stations, but accept new stations if necessary. If possible, only 1 station should comprise a solution. However, if no solution is found with 1 station, then 2 stations may be accepted.
Water Quality Stations NASQAN (Ficke and Hawkinson, 1975) and Benchmark stations (Lawrence, 1987) are matched to a specific USGS station. NAWQA stations (Hirsch and others, 1988) must have a streamgaging station on the same reach.
1These criteria are not necessarily the same as described in USGS, 1998. Some criteria have been added and others modified to reflect better insights or advances in modeling.

Example Costs and Benefits

Costs fall into two classes: (1) station costs or the relative costs of installing and operating a particular type of streamgaging station, and (2) “location” costs that actually are a penalty assigned to a set of stations that satisfies a goal in a less desirable manner. Estimating streamflow from an upstream station, for example, might be less desirable than having a station exactly on the reach where a streamflow estimates is required, and this could be expressed as a cost penalty.

A detailed solution would require assigning an accurate cost to each station. To illustrate the solution technique, an arbitrary relative cost of 10 was assigned to active stations, and 15 to inactive stations that could be reactivated. New stations were assigned a cost of 20. The important point is not absolute costs, but relative costs: with these relative values, using an active station is preferable to reactivating a station, but reactivating a single station is better than using two active stations. Location costs were arbitrarily assigned a value much lower than that of station costs, so that the location cost broke ties among solutions having equal station costs. The location cost depended upon the placement of stations relative to the goal that needs to be satisfied (table 2).

Table 2. List of relative location costs assigned to solutions.
Location of solutionLocation cost
Exact match to USGS station required by the goal 0.0
On the same reach as the goal requires 0.1
One or more stations upstream of the reach the goal requires 0.2
One or more stations downstream of the reach the goal requires 0.3
For a regionalization goal, the station is not an HCDN station (Slack and Landwehr, 1992) 0.1

Because the intent of this example run was to satisfy all goals, the goals were assigned equal benefits by making W the unit vector. This way, the algorithm is free to pick the BFIS in any way that minimizes the cost of the stations required.

The analysis tool was used to examine how the set of USGS streamgaging stations that were active in 1996 met each of the Federal goals. Then, starting over with no stations selected, BFIS’s were chosen to fully meet all the goals. Remember that this analysis uses only streamgaging stations operated by the USGS. There are hundreds of additional streamgaging stations operated by local, State, and other Federal agencies. Although these non-USGS streamgaging stations are not included in this study, the USGS plans to incorporate their contribution to the defined Federal streamgaging goals when data on their locations are compiled.

Determining the Example BFIS

Examining each of the goals yielded a total of 20,865 BFIS’s (table 3). The number of BFIS per goal varies widely and depends on how many ways a goal can be satisfied. Only a single specified station can satisfy some goals, such as Compacts and Decrees. Many different combinations can satisfy others.

Table 3. List of the number of goals of each type and the total number of BFIS’s that satisfy each goal type.
Goal typeNo. of goalsNo. of BFIS
Compacts and Decrees120120
Flooding3,11610,469
Water Budget3293,983
Regionalization8024,231
Quality-Impaired Watersheds5331,902
Water Quality Stations123160
TOTAL5,02320,865

Active Stations – Initial Conditions

Currently active stations achieved the levels of goal satisfaction shown in table 4.

Table 4. Percent of each type of Federal goal satisfied by currently active stations.
Goal typePercent Satisfied
Compacts and Decrees100%
Flooding66%
Water Budget57%
Regionalization58%
Quality-Impaired Watersheds71%
Water Quality Stations81%

Programming Implementation

Determining the BFIS and solving the optimization problem were implemented in software using the Perl programming language (Wall and others, 1996). Perl was chosen because its complex data structures could easily represent stream network topology and the relationship of the BFIS to the individual stations. It would be possible to accomplish the same thing with many other languages. The programs were used to solve a very specific problem and were not written in a generalized form for widespread distribution.

Solution for All Goals

Starting the analysis with no stations selected and continuing until all Federal goals listed herein are satisfied, 2,313 active stations, 857 reactivated stations and 887 new stations would be required, for a total network of 4,057 stations. (There are about 4,300 additional active stations in the network that address other goals, but they are not required to meet the Federal goals considered in this analysis.) A graph of the incremental process to select stations (figure 1) shows two inflection points. For the first 1,000 stations, the incremental procedure adds primarily stations that satisfy more than one goal each. The next 2,700 stations added satisfy mostly one goal each (slope is 1:1). Finally, after adding about 3,700 stations, the procedure adds stations that deal with more complex goals that require more than one station per goal.

Graph
Figure 1. Graph of the number of goals satisfied as a function of the number of stations added.

The relative smoothness of the goal curve shown in figure 1 results from all goals having equal benefits. Had benefits been differentially assigned, more inflection points would have resulted as goals were satisfied generally by order of priority. A map of the selected stations (figure 2) shows their geographic distribution.

Map
Figure 2. Map of stations meeting example Federal goals. Active stations are shown in light gray and reactive stations in black.

Is the Solution Optimal?

The algorithm exhibits a preference for selecting stations that solve multiple goals, as indicated by the slope greater than 1 for the first 1000 stations in figure 1. This is what is expected in an optimal or nearly optimal solution. The solution, though very good, is not, however, guaranteed to be optimal. Situations can be constructed where the algorithm will choose a non-optimal path. Table 5 shows a situation where cost differences caused the algorithm to make the non-optimal choice of A and C. The higher cost of choice B caused it to be rejected in the first step, even though, looking further ahead, it is intuitively the best choice for satisfying all 3 goals. A technique that looked ahead 2 steps would have made the optimal choice. Perhaps an n-step technique, where n is the maximum number of goals met by any BFIS, would be optimal.

Table 5. Example of how the algorithm can make a non-optimal choice.
BFISMeets goalsCostBenefit/Cost (Step 1)Benefit/Cost (Step 2)
A1,212 (select)n/a
B1,2,31.61.8750.625
C2,3121 (select)

Whether the non-optimal situation described in Table 5 actually occurs is unknown. One possibility suggested by reviewers was to select smaller parts of the data set and compare the solution to one determined from complete enumeration. Performing this test, however, would require a substantial amount of additional computer programming and could be the subject of another paper.

Earlier Applications

The example shown in this paper should only be regarded as a starting point for discussions, as other reasonable assumptions about goals and costs could be made. In 1998, the U.S. Geological Survey (1998), for example, employed an earlier version of this technique to evaluate how well the existing streamgaging network met selected Federal goals. However, the study was based upon preliminary data, a few different goals, and slightly different rules for satisfying goals and determining solutions. In 1999, the U.S. Geological Survey (1999) employed this technique as a starting point, but made simplifying assumptions about goals for water quality and long-term change detection.

Conclusions

This paper describes and demonstrates an incremental technique for selecting a nearly optimal set of streamgaging stations to meet any given set of goals for streamflow information, such as flood protection, water allocations, water quality, and long-term changes. Data sets are available that have adequate resolution to apply this technique on a scale suitable for the conterminous United States for Federal goals concerning: compacts and decrees; flooding; water budgets; regionalization of streamflow characteristics; quality-impaired watersheds; and USGS water-quality stations. The technique is sufficiently scalable to deal successfully with the entire network of USGS streamgaging stations.

A preliminary model run indicates that adding 887 new streamgaging stations and reactivating 857 others could meet some of the most important Federal goals for streamgaging in the conterminous United States. This should only be regarded as a starting point for discussions, as other reasonable assumptions about goals and costs could be made. The analysis technique provides a tool for quantitatively evaluating the number and location of streamgaging stations for any list of Federal streamgaging goals, now, or in a future time with different priorities.

Literature Cited

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Burn, D. H., and Goulter, I.C., 1991, An approach to the rationalization of streamflow data collection networks: Journal of Hydrology, v. 122, p. 71-91.

Ficke, J. F., and Hawkinson, R. O., 1975, The National Stream Quality Accounting Network (NASQAN) – Some questions and answers: U.S. Geological Survey Circular 719, 23 p.

Fiering, M. B., 1965, An optimization scheme for gaging: Water Resources Research, v. 1, no. 4, p. 463-470.

Hirsch, R. M., Alley, W. M., and Wilber, W. G., 1988, Concepts for a National Water-Quality Assessment Program: U.S. Geological Survey Circular 1021, 42 p.

Karasev, I. F., 1968, Principles for distribution and prospects for development of a hydrologic network: English version in: Soviet Hydrology, v. 6, p. 560-588.

Lawrence, C. L., 1987, Streamflow characteristics at hydrologic bench-mark stations: U.S. Geological Survey Circular 941, 123 p.

Maddock, T., IV, 1974, An optimum reduction of gages to meet data program constraints: Bulletin of the International Association of Hydrological Sciences, v. 19, no. 3, p. 337-345, 1974.

Moss, M. E., Gilroy, E. J., Tasker, G. D., and Karlinger, M. R., 1982, Design of surface-water data networks for regional information: U. S. Geological Survey Water-Supply Paper 2178, 33p.

Moss, M. E., and Karlinger, M. R., 1974, Surface water network design by regression analysis simulation: Water Resources Research, v. 10, no.3, p. 427-433.

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Slack, J. R., and Landwehr, J. M., 1992, A U.S. Geological Survey Streamflow Data Set for the United States for the Study of Climate Variations, 1874-1988: U.S. Geological Survey Open-File Report 92-129, Compact Disk.

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Thomas, W.O., and Wahl, K.L., 1993, Summary of the nationwide analysis of the cost effectiveness of the U.S. Geological Survey stream-gaging program (1983-88): U.S. Geological Survey Water-Resources Investigations Report 93-4168, 27p.

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